# Lower bound $\sum\limits_{n=1}^\infty a_n/(a_n -1)$ in terms of $a_1$, where $a_n$ is an infinite sequence under some conditions.

This is a question related to a previous question ($a_n$ is an infinite sequence with $\sum\limits_{n=1}^\infty a_n\leq1$ and $0\leq a_n<1$. Prove that $\sum\limits_{n=1}^\infty a_n/(a_n-1)$ converges?).

This time, I want to give a lower bound on the following infinite sum. $$a_n$$ is an infinite sequence with $$\sum\limits_{n=1}^\infty a_n\leq1$$ and $$0\leq a_n<1$$. W.l.o.g., we sort the $$a_n$$ in descending order, and set $$a_1$$ equal to $$0 < c < 1$$. Is it possible to give a lower bound on the sum in terms of $$c$$ on $$\sum\limits_{n=1}^\infty \frac{a_n}{a_n-1}$$?

I tried Taylor series, writing the sum in different forms; Jensen's inequaly is the wrong way around...

• try using en.wikipedia.org/wiki/… – Jneven Mar 15 at 14:47
• $-\frac{1}{1-c}$ will do since $1-a_n \geq 1-c$ and the original sum is positive but $\leq 1$ – Conrad Mar 15 at 14:56
• Thank you, I didn't realise that it was that simple! – Tchaikovski Mar 15 at 15:10